DOCSLIB.ORG

Inflation and De Sitter Thermodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Inflation and De Sitter Thermodynamics CITA-2002-46, hep-th/0212327 Inflation and de Sitter Thermodynamics Andrei Frolov and Lev Kofman Canadian Institute for Theoretical Astrophysics, University of Toronto Toronto, ON, M5S 3H8, Canada (October 22, 2018) Abstract We consider the quasi-de Sitter geometry of the inflationary universe. We cal- culate the energy flux of the slowly rolling background scalar field through the quasi-de Sitter apparent horizon and set it equal to the change of the entropy (1/4 of the area) multiplied by the temperature, dE = T dS. Remarkably, this thermodynamic law reproduces the Friedmann equation for the rolling scalar field. The flux of the slowly rolling field through the horizon of the quasi-de Sitter geometry is similar to the accretion of a rolling scalar field onto a black hole, which we also analyze. Next we add inflaton fluctuations which generate scalar metric perturbations. Metric perturbations result in a variation of the area entropy. Again, the equation dE = T dS with fluctua- tions reproduces the linearized Einstein equations. In this picture as long as the Einstein equations hold, holography does not put limits on the quantum field theory during inflation. Due to the accumulating metric perturbations, the horizon area during inflation randomly wiggles with dispersion increasing with time. We discuss this in connection with the stochastic decsription of inflation. We also address the issue of the instability of inflaton fluctuations in the “hot tin can” picture of de Sitter horizon. arXiv:hep-th/0212327v1 30 Dec 2002 Typeset using REVTEX 1 I. INTRODUCTION The inflationary paradigm established during the last 20 years assumes that the primor- dial equation of state is almost vacuum-like: p ǫ. To realize this equation of state, most models deal with a scalar field φ(t) (or other fields≈ − which in combination act as an effective scalar field) slowly rolling to the minimum of its potential V (φ). During the slow roll regime the homogeneous scalar field produces geometry which can be well approximated by the quasi-de Sitter metric. The full pure de Sitter spacetime, which corresponds to a 4d hyperboloid of constant curvature, can be compactly represented by its Penrose diagram, given by the full square in Fig. 1. It can be covered by different coordinates. Cosmologists most often use coordinates in which the metric is time-dependent and corresponds to an expanding flat universe ds2 = dt2 + e2Ht dr2 + r2dΩ2 , (1) − where dΩ2 = dθ2+sin2 θdφ2. This coordinate system covers the upper half of the hyperboloid, which corresponds to the expansion branch. The Penrose diagram of de Sitter spacetime in flat FRW coordinates is shown on the left panel of Fig. 1. Quasi-de Sitter geometry is dtH(t) described by the scale factor a(t) = a0 e , where the Hubble parameter H is a slowly varying function of time, H˙ H2. R ≪ The time-dependent form of the metric (1) is very convenient for investigating the dy- namics of a scalar field with the equation ✷φ = V,φ (2) and for quantizing this field in the de Sitter spacetime [1]. Among quantum scalar fields with mass m and conformal coupling ξ in de Sitter geometry, the case of minimal coupling ξ =0 and very small mass m H plays an especially important role. Indeed, the regularized vacuum expectation value≪ is δφ2 = 3H4 . Formally, as was noted before the discovery of h i 8π2m2 inflation, this is an odd case since its eigen-spectrum contains an infrared divergent term: δφ2 as m 0. On the other hand, this is the most interesting case for application h i→∞ → t=+ 8 r=const horizon S2 τ=+ 8 t=const R=1/H R=const r=0 R=0 t=- 8 τ=const τ=- 8 FIG. 1. Penrose diagram of de Sitter spacetime in the flat FRW coordinates (left) and the static coordinates (right). Each point represent a sphere S2. Its radius at the horizon (dashed line 1 on the left, edge of diamond on the right) is equal to H . 2 to inflation, since the theory of inflaton (as well as tensor) fluctuations is reduced exactly to this case. Following the time evolution of individual fluctuations, it was found that the infrared divergence can be interpreted as the instability of quantum fluctuations of a very light scalar field, which are accumulated with time δφ2 = H3 t [2,3,4]. Fluctuations of h i 4π δφ induce scalar metric perturbations [5,4,6,7,8]. This picture is a basis of the inflationary paradigm so successfully confirmed observationally. Notice that heavy or conformal fields are not produced by inflation. Further, backreaction of fluctuations δφ leads to the picture of stochastic evolution of quasi-de Sitter geometry [9,10], and at large values of H even to self-reproduction (eternal) of the inflationary universe [11]. Scalar field in the eternal inflationary universe is described naturally in terms of the probability distribution function P (φ, t) [10,12]. Recently, de Sitter spacetime and inflation have drawn significant attention in the theo- retical physics/superstring community. Some of the most interesting topics are holography and the thermodynamics associated with the de Sitter horizon. In this context, the static form of the metric of the de Sitter spacetime dR2 ds2 = (1 H2 R2)dτ 2 + + R2dΩ2 (3) − − (1 H2 R2) − is commonly used. The Penrose diagram of de Sitter spacetime in static coordinates is plotted on the right panel of Fig. 1. The classical result of [13] is that observer at the origin 1 detects a thermal radiation from the de Sitter horizon at R = H with the temperature H 4π T = 2π , and the horizon area A = H2 is associated with the huge (geometrical) entropy A S = 4G . Thermal vacuum in the causal patch (“hot tin can”) corresponds to the Bunch- Davies vacuum of the metric (1) [14,15] and gives a complementary picture of scalar field(s) fluctuations. It is not clear to us, however, how quantum fluctuations in the “hot tin can” picture correspond to the instability of quantum inflaton fluctuations δφ and generation of metric perturbations. We will return to this point at the end of the paper. One of the issues in the holographic approach is the bookkeeping of entropy of de Sitter spacetime. The holography bound declares that the geometrical entropy of the horizon exceeds the entropy of quantum states (of fields and particles) within the volume surrounded by the horizon. It was recently claimed that counting the entropy of quantum fluctuations generated during inflation in the “hot tin can” and comparing it to the change of the apparent horizon entropy violates the holography bound unless an ultra-violet cutoff of order of 1016 ∼ GeV in the momenta of fluctuations is imposed [16]. While it is expected that the approaches based on the time-dependent form of the de Sitter metric with unstable fluctuations and the static form of the de Sitter metric with thermal flux should give us complementary insights, their languages are apparently different. This is partly due to the difference between quasi-de Sitter and pure de Sitter geometries, and partly because different questions are addressed. However, we have to understand how these two different approaches to (quasi-)de Sitter geometry with a scalar field are compatible with each other with respect to such important issues as the generation of fluctuations, entropy and global geometry. In this paper we consider a particular question of how the apparent horizon area A, A or the entropy S = 4G , vary due to the slow roll of the background scalar field and the generation of scalar metric perturbations during inflation. A novel element here is that we 3 combine the concepts of a dynamical, slowly rolling background field and the instability of its fluctuations, with the concept of geometrical, holographic entropy. In Section II, we will calculate a variation of the geometrical entropy due to the energy flux through the apparent horizon area. We find that, remarkably, the thermodynamical relation δE = TdS is equivalent to the Einstein equation for the rolling inflaton field. In a sense, our derivation of a correspondence between thermodynamics and the Einstein equations for inflation is a realization of such a correspondence found in an inspiring paper [17] for local accelerating observers. However, we introduce a technique to treat the apparent horizon of R S2 topology which is different from the description [17] of a local Rindler horizon for an× accelerating observer. As we will see, a non-vanishing flux is generated by the kinetic term φ˙2 of the slowly rolling inflaton field. It turns out that this problem is very similar to the problem of the interaction of a homogeneous rolling scalar field with a runaway potential V (φ) and a black hole. In Section III, we switch our attention from inflation to black holes. A rolling scalar field interacting with a black hole is a transparent illustration of the energy flow of a light scalar field through a horizon. In Section IV, we return to inflation. On top of the rolling background inflaton, we con- sider inflaton fluctuations δφ, which generate scalar metric perturbations Φ. We study the energy flux through the horizon including inhomogeneous δφ fluctuations and corresponding variations in the area of the horizon, or entropy dS, which are sensitive to the scalar metric perturbations Φ. In this case, the calculations are more involved than the calculations for the homogeneous time dependent background field in Section II. This happens because there is no exact Killing vector generating the horizon. However, for metric perturbations which preserve spherical symmetry we still can define TdS and compare it with the energy flux through the horizon.
Load more

Recommended publications
  • The State of the Multiverse: the String Landscape, the Cosmological Constant, and the Arrow of Time
    The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time Raphael Bousso Center for Theoretical Physics University of California, Berkeley Stephen Hawking: 70th Birthday Conference Cambridge, 6 January 2011 RB & Polchinski, hep-th/0004134; RB, arXiv:1112.3341 The Cosmological Constant Problem The Landscape of String Theory Cosmology: Eternal inflation and the Multiverse The Observed Arrow of Time The Arrow of Time in Monovacuous Theories A Landscape with Two Vacua A Landscape with Four Vacua The String Landscape Magnitude of contributions to the vacuum energy graviton (a) (b) I Vacuum fluctuations: SUSY cutoff: ! 10−64; Planck scale cutoff: ! 1 I Effective potentials for scalars: Electroweak symmetry breaking lowers Λ by approximately (200 GeV)4 ≈ 10−67. The cosmological constant problem −121 I Each known contribution is much larger than 10 (the observational upper bound on jΛj known for decades) I Different contributions can cancel against each other or against ΛEinstein. I But why would they do so to a precision better than 10−121? Why is the vacuum energy so small? 6= 0 Why is the energy of the vacuum so small, and why is it comparable to the matter density in the present era? Recent observations Supernovae/CMB/ Large Scale Structure: Λ ≈ 0:4 × 10−121 Recent observations Supernovae/CMB/ Large Scale Structure: Λ ≈ 0:4 × 10−121 6= 0 Why is the energy of the vacuum so small, and why is it comparable to the matter density in the present era? The Cosmological Constant Problem The Landscape of String Theory Cosmology: Eternal inflation and the Multiverse The Observed Arrow of Time The Arrow of Time in Monovacuous Theories A Landscape with Two Vacua A Landscape with Four Vacua The String Landscape Many ways to make empty space Topology and combinatorics RB & Polchinski (2000) I A six-dimensional manifold contains hundreds of topological cycles.
    [Show full text]
  • On Asymptotically De Sitter Space: Entropy & Moduli Dynamics
    On Asymptotically de Sitter Space: Entropy & Moduli Dynamics To Appear... [1206.nnnn] Mahdi Torabian International Centre for Theoretical Physics, Trieste String Phenomenology Workshop 2012 Isaac Newton Institute , CAMBRIDGE 1 Tuesday, June 26, 12 WMAP 7-year Cosmological Interpretation 3 TABLE 1 The state of Summarythe ofart the cosmological of Observational parameters of ΛCDM modela Cosmology b c Class Parameter WMAP 7-year ML WMAP+BAO+H0 ML WMAP 7-year Mean WMAP+BAO+H0 Mean 2 +0.056 Primary 100Ωbh 2.227 2.253 2.249−0.057 2.255 ± 0.054 2 Ωch 0.1116 0.1122 0.1120 ± 0.0056 0.1126 ± 0.0036 +0.030 ΩΛ 0.729 0.728 0.727−0.029 0.725 ± 0.016 ns 0.966 0.967 0.967 ± 0.014 0.968 ± 0.012 τ 0.085 0.085 0.088 ± 0.015 0.088 ± 0.014 2 d −9 −9 −9 −9 ∆R(k0) 2.42 × 10 2.42 × 10 (2.43 ± 0.11) × 10 (2.430 ± 0.091) × 10 +0.030 Derived σ8 0.809 0.810 0.811−0.031 0.816 ± 0.024 H0 70.3km/s/Mpc 70.4km/s/Mpc 70.4 ± 2.5km/s/Mpc 70.2 ± 1.4km/s/Mpc Ωb 0.0451 0.0455 0.0455 ± 0.0028 0.0458 ± 0.0016 Ωc 0.226 0.226 0.228 ± 0.027 0.229 ± 0.015 2 +0.0056 Ωmh 0.1338 0.1347 0.1345−0.0055 0.1352 ± 0.0036 e zreion 10.4 10.3 10.6 ± 1.210.6 ± 1.2 f t0 13.79 Gyr 13.76 Gyr 13.77 ± 0.13 Gyr 13.76 ± 0.11 Gyr a The parameters listed here are derived using the RECFAST 1.5 and version 4.1 of the WMAP[WMAP-7likelihood 1001.4538] code.
    [Show full text]
  • University of California Santa Cruz Quantum
    UNIVERSITY OF CALIFORNIA SANTA CRUZ QUANTUM GRAVITY AND COSMOLOGY A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS by Lorenzo Mannelli September 2005 The Dissertation of Lorenzo Mannelli is approved: Professor Tom Banks, Chair Professor Michael Dine Professor Anthony Aguirre Lisa C. Sloan Vice Provost and Dean of Graduate Studies °c 2005 Lorenzo Mannelli Contents List of Figures vi Abstract vii Dedication viii Acknowledgments ix I The Holographic Principle 1 1 Introduction 2 2 Entropy Bounds for Black Holes 6 2.1 Black Holes Thermodynamics ........................ 6 2.1.1 Area Theorem ............................ 7 2.1.2 No-hair Theorem ........................... 7 2.2 Bekenstein Entropy and the Generalized Second Law ........... 8 2.2.1 Hawking Radiation .......................... 10 2.2.2 Bekenstein Bound: Geroch Process . 12 2.2.3 Spherical Entropy Bound: Susskind Process . 12 2.2.4 Relation to the Bekenstein Bound . 13 3 Degrees of Freedom and Entropy 15 3.1 Degrees of Freedom .............................. 15 3.1.1 Fundamental System ......................... 16 3.2 Complexity According to Local Field Theory . 16 3.3 Complexity According to the Spherical Entropy Bound . 18 3.4 Why Local Field Theory Gives the Wrong Answer . 19 4 The Covariant Entropy Bound 20 4.1 Light-Sheets .................................. 20 iii 4.1.1 The Raychaudhuri Equation .................... 20 4.1.2 Orthogonal Null Hypersurfaces ................... 24 4.1.3 Light-sheet Selection ......................... 26 4.1.4 Light-sheet Termination ....................... 28 4.2 Entropy on a Light-Sheet .......................... 29 4.3 Formulation of the Covariant Entropy Bound . 30 5 Quantum Field Theory in Curved Spacetime 32 5.1 Scalar Field Quantization .........................
    [Show full text]
  • On the Limits of Effective Quantum Field Theory
    RUNHETC-2019-15 On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Abstract We recapitulate multiple arguments that Eternal Inflation and the String Landscape are actually part of the Swampland: ideas in Effective Quantum Field Theory that do not have a counterpart in genuine models of Quantum Gravity. 1 Introduction Most of the arguments and results in this paper are old, dating back a decade, and very little of what is written here has not been published previously, or presented in talks. I was motivated to write this note after spending two weeks at the Vacuum Energy and Electroweak Scale workshop at KITP in Santa Barbara. There I found a whole new generation of effective field theorists recycling tired ideas from the 1980s about the use of effective field theory in gravitational contexts. These were ideas that I once believed in, but since the beginning of the 21st century my work in string theory and the dynamics of black holes, convinced me that they arXiv:1910.12817v2 [hep-th] 6 Nov 2019 were wrong. I wrote and lectured about this extensively in the first decade of the century, but apparently those arguments have not been accepted, and effective field theorists have concluded that the main lesson from string theory is that there is a vast landscape of meta-stable states in the theory of quantum gravity, connected by tunneling transitions in the manner envisioned by effective field theorists in the 1980s.
    [Show full text]
  • Geometric Origin of Coincidences and Hierarchies in the Landscape
    Geometric origin of coincidences and hierarchies in the landscape The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bousso, Raphael et al. “Geometric origin of coincidences and hierarchies in the landscape.” Physical Review D 84.8 (2011): n. pag. Web. 26 Jan. 2012. © 2011 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.84.083517 Publisher American Physical Society (APS) Version Final published version Citable link http://hdl.handle.net/1721.1/68666 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 84, 083517 (2011) Geometric origin of coincidences and hierarchies in the landscape Raphael Bousso,1,2 Ben Freivogel,3 Stefan Leichenauer,1,2 and Vladimir Rosenhaus1,2 1Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720-7300, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720-8162, USA 3Center for Theoretical Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 2 August 2011; published 14 October 2011) We show that the geometry of cutoffs on eternal inflation strongly constrains predictions for the time scales of vacuum domination, curvature domination, and observation. We consider three measure proposals: the causal patch, the fat geodesic, and the apparent horizon cutoff, which is introduced here for the first time. We impose neither anthropic requirements nor restrictions on landscape vacua. For vacua with positive cosmological constant, all three measures predict the double coincidence that most observers live at the onset of vacuum domination and just before the onset of curvature domination.
    [Show full text]
  • Arxiv:Hep-Th/0209231 V1 26 Sep 2002 Tbs,Daai N Uigi Eurdfrsae Te Hntetemlvcu Olead to Vacuum Thermal the Anisotropy
    SLAC-PUB-9533 BRX TH-505 October 2002 SU-ITP-02/02 Initial conditions for inflation Nemanja Kaloper1;2, Matthew Kleban1, Albion Lawrence1;3;4, Stephen Shenker1 and Leonard Susskind1 1Department of Physics, Stanford University, Stanford, CA 94305 2Department of Physics, University of California, Davis, CA 95616 3Brandeis University Physics Department, MS 057, POB 549110, Waltham, MA 02454y 4SLAC Theory Group, MS 81, 2575 Sand Hill Road, Menlo Park, CA 94025 Free scalar fields in de Sitter space have a one-parameter family of states invariant under the de Sitter group, including the standard thermal vacuum. We show that, except for the thermal vacuum, these states are unphysical when gravitational interactions are arXiv:hep-th/0209231 v1 26 Sep 2002 included. We apply these observations to the quantum state of the inflaton, and find that at best, dramatic fine tuning is required for states other than the thermal vacuum to lead to observable features in the CMBR anisotropy. y Present and permanent address. *Work supported in part by Department of Energy Contract DE-AC03-76SF00515. 1. Introduction In inflationary cosmology, cosmic microwave background (CMB) data place a tanta- lizing upper bound on the vacuum energy density during the inflationary epoch: 4 V M 4 1016 GeV : (1:1) ∼ GUT ∼ Here MGUT is the \unification scale" in supersymmetric grand unified models, as predicted by the running of the observed strong, weak and electromagnetic couplings above 1 T eV in the minimal supersymmetric standard model. If this upper bound is close to the truth, the vacuum energy can be measured directly with detectors sensitive to the polarization of the CMBR.
    [Show full text]
  • Geometric Approaches to Quantum Field Theory
    GEOMETRIC APPROACHES TO QUANTUM FIELD THEORY A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Kieran T. O. Finn School of Physics and Astronomy Supervised by Professor Apostolos Pilaftsis BLANK PAGE 2 Contents Abstract 7 Declaration 9 Copyright 11 Acknowledgements 13 Publications by the Author 15 1 Introduction 19 1.1 Unit Independence . 20 1.2 Reparametrisation Invariance in Quantum Field Theories . 24 1.3 Example: Complex Scalar Field . 25 1.4 Outline . 31 1.5 Conventions . 34 2 Field Space Covariance 35 2.1 Riemannian Geometry . 35 2.1.1 Manifolds . 35 2.1.2 Tensors . 36 2.1.3 Connections and the Covariant Derivative . 37 2.1.4 Distances on the Manifold . 38 2.1.5 Curvature of a Manifold . 39 2.1.6 Local Normal Coordinates and the Vielbein Formalism 41 2.1.7 Submanifolds and Induced Metrics . 42 2.1.8 The Geodesic Equation . 42 2.1.9 Isometries . 43 2.2 The Field Space . 44 2.2.1 Interpretation of the Field Space . 48 3 2.3 The Configuration Space . 50 2.4 Parametrisation Dependence of Standard Approaches to Quan- tum Field Theory . 52 2.4.1 Feynman Diagrams . 53 2.4.2 The Effective Action . 56 2.5 Covariant Approaches to Quantum Field Theory . 59 2.5.1 Covariant Feynman Diagrams . 59 2.5.2 The Vilkovisky–DeWitt Effective Action . 62 2.6 Example: Complex Scalar Field . 66 3 Frame Covariance in Quantum Gravity 69 3.1 The Cosmological Frame Problem .
    [Show full text]
  • Observational Cosmology - 30H Course 218.163.109.230 Et Al
    Observational cosmology - 30h course 218.163.109.230 et al. (2004–2014) PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Thu, 31 Oct 2013 03:42:03 UTC Contents Articles Observational cosmology 1 Observations: expansion, nucleosynthesis, CMB 5 Redshift 5 Hubble's law 19 Metric expansion of space 29 Big Bang nucleosynthesis 41 Cosmic microwave background 47 Hot big bang model 58 Friedmann equations 58 Friedmann–Lemaître–Robertson–Walker metric 62 Distance measures (cosmology) 68 Observations: up to 10 Gpc/h 71 Observable universe 71 Structure formation 82 Galaxy formation and evolution 88 Quasar 93 Active galactic nucleus 99 Galaxy filament 106 Phenomenological model: LambdaCDM + MOND 111 Lambda-CDM model 111 Inflation (cosmology) 116 Modified Newtonian dynamics 129 Towards a physical model 137 Shape of the universe 137 Inhomogeneous cosmology 143 Back-reaction 144 References Article Sources and Contributors 145 Image Sources, Licenses and Contributors 148 Article Licenses License 150 Observational cosmology 1 Observational cosmology Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors. Early observations The science of physical cosmology as it is practiced today had its subject material defined in the years following the Shapley-Curtis debate when it was determined that the universe had a larger scale than the Milky Way galaxy. This was precipitated by observations that established the size and the dynamics of the cosmos that could be explained by Einstein's General Theory of Relativity.
    [Show full text]
  • Arxiv:1707.01092V2 [Gr-Qc]
    September 2017 Holographic Entanglement Entropy and Cyclic Cosmology Paul H. Frampton∗ Dipartimento di Matematica e Fisica ”Ennio De Giorgi” Universit`adel Salento and INFN Lecce, Via Arnesano 73100 Lecce,Italy. Abstract We discuss a cyclic cosmology in which the visible universe, or introverse, is all that is accessible to an observer while the extroverse represents the total spacetime originating from the time when the dark energy began to dominate. It is argued that entanglement entropy of the introverse is the more appropriate quantity to render infinitely cyclic, rather than the entropy of the total universe. Since vanishing entanglement entropy implies disconnected spacetimes, at the turnaround when the introverse entropy is zero the disconnected extroverse can be jettisoned with impunity. arXiv:1707.01092v2 [gr-qc] 1 Sep 2017 ∗email: [email protected] 1 Introduction Cyclic cosmology with an infinite number of cycles which include – expansion, turnaround, contraction, bounce – is an appealing theory, and one of the most recent attempts at a solution was in [1]. As emphasized first by Tolman [2,3], entropy and the second law of thermodynamics provide a significant challenge to implementing a cyclic theory. The approach was based on the so-called Come Back Empty (CBE) principle which chooses the turnaround as the point in the cycles where entropy is jettisoned. A successful universe must be selected with zero entropy at the turnaround to begin an adiabatic contraction to the bounce while empty of all matter including black holes. In this article we provide a better understanding of CBE based on the holographic principle [4–6].
    [Show full text]
  • Ads/CFT and Black Holes - Lecture 3
    2230-3 Spring School on Superstring Theory and Related Topics 28 March - 5 April, 2011 AdS/CFT and Black Holes - Lecture 3 A. MALONEY McGill University Montreal Canada A de Sitter Farey Tail ICTP Alex Maloney, McGill University Castro, Lashkari & A. M. Overview The Problem Quantum cosmology is confusing: What are the appropriate observables for eternal inflation? What is the meaning and origin of the entropy of a cosmological horizon? Is quantum mechanics the right language to describe cosmology? We were able to answer analogous questions about black hole physics using AdS/CFT. Let us be bold and apply the same techniques to cosmology. de Sitter Space de Sitter space is the maximally symmetric solution of general relativity with a positive cosmological constant. √ 1 − − 2 S(g)= g R 2 G M The universe inflates eternally. The three dimensional theory can be solved exactly, in the sense that the partition function − [ ] Z = Dge S g can be computed exactly. We will be inspired by AdS/CFT but we will not use it. The Idea The saddle point approximation is 1 −S[g] −kS0+S1+ S2+... Z = Dge = e k g0 where k = /G is the coupling. The approximation becomes exact if we can Find all classical saddles Compute all perturbative corrections around each saddle We can do it! The new classical saddles have a straightforward physical interpretation and lead to quantum gravitational effects for de Sitter observers. The Plan for Today: • The de Sitter Farey Tail • The Partition Function • Discussion Causal Structure of de Sitter Eternal inflation is the ultimate existential nightmare.
    [Show full text]
  • Xavier Calmet Editor Quantum Aspects of Black Holes Fundamental Theories of Physics
    Fundamental Theories of Physics 178 Xavier Calmet Editor Quantum Aspects of Black Holes Fundamental Theories of Physics Volume 178 Series editors Henk van Beijeren Philippe Blanchard Paul Busch Bob Coecke Dennis Dieks Detlef Dürr Roman Frigg Christopher Fuchs Giancarlo Ghirardi Domenico J.W. Giulini Gregg Jaeger Claus Kiefer Nicolaas P. Landsman Christian Maes Hermann Nicolai Vesselin Petkov Alwyn van der Merwe Rainer Verch R.F. Werner Christian Wuthrich More information about this series at http://www.springer.com/series/6001 Xavier Calmet Editor Quantum Aspects of Black Holes 123 Editor Xavier Calmet Department of Physics and Astronomy University of Sussex Brighton UK ISBN 978-3-319-10851-3 ISBN 978-3-319-10852-0 (eBook) DOI 10.1007/978-3-319-10852-0 Library of Congress Control Number: 2014951685 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
    [Show full text]
  • Arxiv:1511.02114V5
    Hawking modes and the optimal disperser : Holographic lessons from the observer’s causal-patch unitarity J. Koohbor (a) ∗, M. Nouri-Zonoz (a)† and A. Tavanfar (b)(c) ‡ (a): Department of Physics, University of Tehran, North Karegar Ave., Tehran 14395-547, Iran. (b) : Instituto de Telecomunica¸co˜es, Physics of Information and Quantum Technologies Group, Lisbon, Portugal. (c): International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil. Abstract Based on an observer-centric methodology, we pinpoint the basic origin of the spectral Planckian- ity of the asymptotic Hawking modes in the conventional treatments of the evaporating horizons. By considering an observer who analyzes a causal horizon in a generic spacetime, we first clarify how the asymptotic Planckian spectrum is imposed on the exponentially redshifted Hawking modes through a geometric dispersion mechanism developed by a semiclassical environment which is com- posed by all the modes that build up the curvature of the causal patch of the asymptotic observer. We also discuss the actual microscopic phenomenon of the Hawking evaporation of generic causal horizons. Our quantum description is based on a novel holographic scheme of gravitational open quantum systems in which the degrees of freedom that build up the curvature of the observer’s causal patch interact with the radiated Hawking modes, initially as environmental quanta, and after a crossover time, as quantum defects. Planckian dispersion of the modes would only be de- arXiv:1511.02114v5 [hep-th] 24 Apr 2018 veloped in the strict thermodynamic limit of this quantum environment, called optimal disperser, which is nevertheless avoided holographically.
    [Show full text]